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Retaining Walls
UNIT 9 RETAINING WALLS
Structure 9.1 Introduction
Objectives
9.2 Pressure on Retaining Walls Due to Liquids 9.3 Analysis of a Masonry Dam 9.4 Stability of Retaining Walls 9.5 Pressure on Retaining Walls Due to Granular Materials : Rankine’s
Theory 9.6 Calculation of Pressures at the Base of Retaining Wall 9.7 Retaining Walls with Inclined Earth Slopes 9.8 Minimum Depth of Foundation 9.9 Summary 9.10 Answers to SAQs
9.1 INTRODUCTION When a mass of material, whether liquid or granular solid, is required to be maintained at a higher level on only one side, the dividing structure between the two is known as a ‘retaining wall’ when a solid is retained, or a ‘dam’ when a liquid is retained.
Retaining Wall
Earth Dam
Water H
h
u/s Side
d/s Side
Free Board
hf
(a) (b) Figure 9.1
In Figure 9.1(a), water at a higher level, as in a reservoir, etc., is maintained at a height ‘h’ on the right hand (up-stream) side of the dam. In Figure 9.1(b) earth on the right hand side is retained at a level higher than on the left hand (down-stream) side, and it is maintained so by the wall. In either case the dam or wall acts as a retaining wall which prevents the material (liquid or granular solid) from slipping or flowing towards the left; and, therefore, it is under a horizontal pressure which tends to overturn it or move it towards the left. The retaining wall should be good enough to prevent any such event or fail under such a pressure. The weight of the wall and the horizontal pressure exerts pressure on the soil below foundation which should also be within safe limits. Objectives After studying this unit, you should be able to
• calculate the pressures acting on a retaining wall due to the retained fluid or granular solid,
214
Theory of Structures-I • design a wall which is strong enough to withstand these pressures, • proportion a wall so that no overturning or sliding takes place, and • calculate the soil pressures below the wall base and see that pressures
are within safe limits.
9.2 PRESSURE ON RETAINING WALLS DUE TO LIQUIDS
In Figure 9.2(a), the face of the wall in contact with the liquid is vertical, hence by the well known law of hydrostatics, the fluid pressure at any point A on the wall at a depth h is wh, where w is the density (or unit weight) of the liquid, and acts normal to the wall, (i.e. in the horizontal direction). The pressure diagram will be a triangle with zero pressure at top and a pressure wH at the bottom of the wall. The resultant force per unit length of wall will be the area of the pressure
triangle, i.e. 2
2wHP = and it will act at the CG of the triangle, i.e. at a height
3H from the base of the wall.
Water
(a)
(b) Figure 9.2
In Figure 9.2(b), the face of the wall is inclined at an angle α with the vertical, hence the pressure at a depth h is wh and acts normal to the wall, i.e. at an angle α with the horizontal. The pressure diagram is again triangular, and the resultant
2
2wHP = inclined at α with the horizontal. The horizontal component of P will
be 2
2wH cos α acting at depth 3H from the bottom.
9.3 ANALYSIS OF A MASONRY DAM
H
h
p = wH
p = wh
αP = wH2/2
α H/3
α
90o α
p = wH2/2
H
h
Ap = wh
H/3
p = wH
215
Retaining WallsFigure 9.3 shows a masonry dam of trapezoidal cross section ABCD with water
face BC vertical. The top width of the dam is ‘a’ and its base width ‘b’ (b > a). The height of the dam is H, and the density of the material of the dam (masonry) is ρ. Let the height of water retained be ‘h’ (which is less than H, due to the free-board kept over reservoir water level).
H h
A Ba
h/3
P = wh2/2
Free Board
D CCM N e b/2 b/2
b
w
+ w/b
6We/b2
2ph/b2
(i)
(ii)
(iii)
Heel Toe
−
−
(Comp)
(Tensile)(Comp)
(Tensile)
+
(Comp)
+
(i) Pressure distribution due to vertical load
(ii) Pressure distribution due to moment W . e
(iii) Pressure distribution due to water pressure P
Figure 9.3 : Pressure Distribution below Dam Base
The two forces acting on the dam is its self weight W which is
ρ2
a bW H+= ⋅
and the horizontal pressure of water P as shown. The self weight W does not pass through the middle point M of the base BC, rather it passes through N. It can be easily seen that
2 2
3( )a ab bNC = x
a + b+ +
= . . . (9.1)
Thus, the eccentricity ‘e’ of the weight W is 2 2
2 3( )a ab + bbe = MN = MC NC
a + b+
− = − . . . (9.2)
Considering 1.0 m length of the dam, the pressures acting at the base of the dam can be divided into the following three types
216
(a) Due to the vertical load W, it is A
W (uniformly compressive) Theory of Structures-I
∴ b
WbW
AWp =
×==
11 [Figure 9.3 (i)]
(b) Due to the moment W. e caused by eccentricity of W;
2 23
.6 .2
112
bW . eM . y W ep
I bb
⎛ ⎞⎜ ⎟⎝ ⎠= ± = ± = ±
⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠
. . . (9.3)
There will be compressive force at heel C and tensile at toe D, (Figure 9.3(ii)) as the point N is nearer to C.
(c) Due to the moment 3hP ⋅ caused by the horizontal water pressure
acting at a height 3h from the base
∴ 3 23
.2 Ph3 2
112
h bPMypI bb
⎛ ⎞⎜ ⎟⎝ ⎠= ± = ± = ±⎛ ⎞×⎜ ⎟⎜ ⎟⎝ ⎠
. . . (9.4)
It will cause compression at toe D and tension at heel C as shown in Figure 9.3(iii).
The resultant pressure at any point is the algebraic sum of the pressures p1, p2 and p3 as calculated above.
For the safety of the dam the following conditions have to be ensured :
(a) There should not be any tensile stress in the dam, as masonry is not expected to resist any tensile stress.
(b) The total compressive stresses at any section (at any point) of the dam should not exceed the allowable compressive stress in masonry.
(c) The maximum pressure below the foundation should not exceed the allowable bearing capacity of soil.
(d) The dam should not fail due to lack of stability, i.e. by sliding or by overturning due to the horizontal water pressure.
This last point is discussed in the next section.
9.4 STABILITY OF RETAINING WALLS
Sliding
As retaining walls are subjected to horizontal (as well as vertical loads) there will be a tendency of the wall to move or slide in the direction of the force. This is resisted by the frictional force acting at the base of the wall. If the coefficient of friction between the base and the soil is μ and W is the total vertical forces, then as the vertical reaction R = W, the frictional force
217
Retaining WallsF = μ R = μ W.
Hence, for a condition of no sliding, total horizontal forces
Ph < μ W
h
WFSPμ
= . . . (9.5)
The ratio h
WPμ is known as the factor of safety against sliding, and it should
never be less than 1.0. Normally the factor of safety against sliding is taken as 2.0.
Overturning
Another effect of the horizontal forces is its tendency to overturn the retaining wall about its toe D. The overturning moment is .hP y , where y is the distance of resultant of the horizontal forces above the base.
The stabilizing moment which is caused by the weight of the wall (or any material resting on the slopes of the wall) is given by zW ⋅ , where
)( xbz −= is the distance of the resultant from the toe D.
For the stabilizing moment to be greater than overturning moment, we must
have hW z P y⋅ > ⋅ and the factor of safety against overturning is h
W zP y⋅⋅
which should not be less than 1.0. Normally the safety factor is taken as 2.0.
h
W zFSP y
= . . . (9.6)
h
A B
C D
P
y W
Toe
z x
b Heel
a
PhW
R
μR
218
Figure 9.4 Theory of Structures-I
Example 9.1
A masonry dam 8 m high is 1.5 m wide at the top and 5 m wide at the base. It retains water to a depth of 7.5m. The water face of the dam is vertical. Find the maximum and minimum stresses at the base. The weight of the masonry is 22.4 kN/m3.
Solution
Consider 1 m width of the dam (Figure 9.5).
Figure 9.5
Weight of masonry, 1.5 5.0 8.0 22.4 582.4 kN2
W += × × =
Its line of action meet the base DC at N here NC is 2 2 2 21.5 1.5 5 5 1.782 m
3( ) 3(1.5 5)a ab bNC = x =
a + b+ + + × +
= =+
− 100.359 kN/mm2
+ 100.359 kN/mm2
+ 184.871 kN/m2
D
(ii) Pressure Diagram Du
+ 48.089 kN/m2
C
e to Eccentricity ‘e’
− 168.75 kN/m2
re Diagram
ram Due to W
+ 168.75 kN/m2
(iii) Pressure Diagram Due to P
(iv) Final Pressu
(i) Pressure Diag
+ Compression 116.480 kN/m2
8 m 7.5 m
A 1.5 B
2.5 m
p = 281.25 G
D M e2.5 m
5 m
W = 582.4 kN
CCN2.5 m
1.782 m
219
Retaining Walls∴ Eccentricity, e = MN = MC – x = 2.5 – 1.782 = 0.718 m
The horizontal water pressure P is 22 10 (7.5) 281.25 kN
2 2WhP ×
= = =
and it acts at a height of 7.5 2.5 m3
= above the base DC.
∴ (a) Pressure due to weight W is
1582 4 116 48
5W .p .b
= = = kN/m2
and is uniform at the base DC. (b) Pressure due to the eccentricity of weight W is
22 2 2
6 6 582.4 0.718 100 359 kN/mm5
Wep .b
× ×= ± = ± = ±
It varies from a tensile stress – 100.359 kN/m2 at D to a compressive Stress + 100.359 kN/m2 at C at base DC.
(c) Pressure due to the moment of the horizontal water pressure p is
23 2 2
2 281.25 7.52 168.750 kN/m5
phpb
× ×= ± = ± = ±
which varies from a compressive stress at D to tensile at C. The final pressure diagram is shown in Figure 9.5(iv) as an algebraic sum of p1, p2, p3 and is 184.871 kN/m2 (comp.) at D to 48.089 kN/m2 (comp.) at C.
Example 9.2
Find the factor of safety against (a) sliding and (b) overturning for the dam section in Example 9.1 (coefficient of friction between soil and dam base is 0.6).
Solution (a) The vertical load is the weight (W) of the masonry dam which has
been calculated as 582.4 kN per metre of the wall
The frictional force at base is F = μ W or F = 0.6 × 582.4 kN = 349.44 kN The sliding force is the horizontal pressure P = 281.25 kN.
Hence, the safety factory against sliding 349 44 1 243281 25
. .
.= =
(b) The moment of the vertical load W about the toe D is (W b x− ) = 582.4 (5 – 1.782) = 1874.16 kNm
which is the stabilizing moment. The overturning moment due to water pressure is
3hP ⋅ = 281.25 × 2.5 = 703.125 kNm
220
∴ The factor of safety against overturning 67.2125.70316.1874
== . Theory of Structures-I
Example 9.3
Design the section of a trapezoidal masonry dam (with water face vertical) to impound water up to 29 m depth on the upstream side, with a free-board of 1 m. The maximum allowable pressure on base is 900 kN/m2. Assume no tension in masonry (weight of masonry = 23 kN/m3).
Solution
(Refer Figure 9.6)
Total height of dam = 29 + 1 = 30 m
Let the top width = a metre
Base width = b metre
Considering one metre length of dam.
Figure 9.6
Weight of masonry, ρ2
a + bW H= ⋅
30232
××+
=ba
= 345 (a + b) kN
Maximum water pressure 22 10 29 4205 kN
2 2whP ×
= = =
Maximum pressure at toe D
22 2
6 2 900 kN/mDWeW Php
b b b= − + = (maximum allowable pressure)
Minimum pressure at heel C
2 2
6 2 0CWeW Pp
b b bh
= + − = (for no tension)
Substituting the values of W, P and h in the above equation
A Ba
29/3 m
1 m (Free Board)
30 m 29 mO
/2
w = p
D CM N eb/2 b
b
(a+b/2)H
Toe Heel
221
Retaining Walls
2 2345 ( ) 6 345( ) 2 4205 29 900a + b a b e
b b b
× + ⋅ × ×− + = . . . (A)
2 2345 ( ) 6 345( ) 2 4205 29 0a + b a b e
b b b
× + ⋅ × ×+ − = . . .
(B) Adding Eqs. (A) and (B), we get
2 345 900,a + bb
⎛ ⎞× =⎜ ⎟⎝ ⎠
giving 1.304a + bb
=
or a = 0.304 b Subtracting Eq. (B) from Eq. (A) we get
2 24 4205 292 6 345 ( ) 900ea + b
b b
× ×× × − =
– 4140 (a + b) e + 487780 = 900 b2 or b2 + 4.6 (a + b) e – 542 = 0 . . . (C) Also we know that
2be + x = [
2DCMN + NC = MC =∵ ]
where 2 2
3( )a ab + bx
a + b+
= . . . (D)
Substituting the value of a 2 2 2(0.304 ) 0.304 0.357
3(0.304 )b b bx b
b b+ +
= =+
∴ 0.5 0.357 0.143be = x b b b2− = − =
Substituting the value of a and e in Eq. (C), we get 2 4.6 (0.304 ) (0.143 ) 542 0b b b b+ + − =
giving b = 17.17 m , say 17.2 m and a = 0.304 b , 5.2 m (say)
Check for Stability
Assuming coefficient of friction μ = 0.6 W = 345 (a + b) = 345 (17 + 5.2) = 7659 kN
∴ μ W = 0.6 × 7659 = 4595.4 kN P = 4205 kN/m2
∴ Factor of safety against sliding
4595.4 1.09 14205
= = >
and factor of safety against overturning
222
Theory of Structures-I
7659 (0.143 0.5) 172 2.06 1294205
3 3
bW e
hP
⎛ ⎞+⎜ ⎟ + ×⎝ ⎠= = =⋅ ×
> ∴ OK.
9.5 PRESSURES ON RETAINING WALLS DUE TO GRANULAR MATERIALS – RANKINE’S THEORY
Earth material (or any other granular material like sand, cement grains, etc.) when collected in a heap tends to adopt a slope, which is characteristics of the material and is known as its “angle of repose”. This is due to the internal friction between the grains of the material. A French engineer, Rankine, developed the theory of earth pressures in the 18th century. Rankine’s earth pressure equation gives the horizontal earth pressure on the vertical face of a retaining wall due to a level earth fill behind if the earth pressure (p) can be expressed as
1 sin kwh1 sin
p wh − φ= =
+ φ . . . (9.7)
where 1 sin1 sin
k − φ=
+ φ is the Rankine’s earth pressure coefficient,
w = Unit weight of the material, h = Depth of the point at which pressure is sought, and
φ = Angle of internal friction (or angle of repose) of the material. This is shown in Figure 9.7 where the maximum pressure on the vertical face of the retaining wall AB at depth H is
1 sin1 sin
p wH − φ=
+ φ . . . (9.8)
and the pressure diagram is a triangle. The total pressure is the area of the triangle and is given as
2 1 sin12 1 sin
P w H − φ=
+ φ . . . (9.9)
and acts at a depth 3H from the bottom
a
A
B
H/3
H
G
h
1 sin.1 sin
p wh − φ=
+ φ
b
2 1 sin12 1 sin
P wH − φ=
+ φ
1 sin.1 sin
wH − φ+ φ
223
Retaining Walls
Figure 9.7
9.6 CALCULATION OF PRESSURES AT THE BASE OF RETAINING WALL
In all these calculations we consider 1 m width of the wall. Here P is the total horizontal pressure acting on a retaining wall and W is the total weight of all vertical loads acting on it including its own weight. If the resultant of these forces meet the base at point E and O is the midpoint of the base then OE = e is the eccentricity of the resultant R.
H
h
A B
CD
Oe
P P
W
b
E
R
(a) Section of Wall
D E O C
1 m
D′ C′O′Plan
(b) Pressure Diagram
Figure 9.8
Therefore, the base will be subjected to an eccentric load caused by a direct vertical load W and a moment M (= Ph ± W.e). The stress at any point distant x from the centre O will be given by
0
.M xWp = ±A I
. . . (9.10)
where b is the width of base DC, and I0 is the moment of inertia of the base
rectangle CC′DD′ about the centre line OO′ 3 3
01Here
12 12b bI
⎛ ⎞×= =⎜ ⎟⎜ ⎟
⎝ ⎠
and A (= 1 × b = b) is area of the rectangle CC′DD′. The extreme pressure (maximum and minimum) acts at the point D and C
where, 2bx = .
Substituting these values in Eq. (9.10), we have
224
Theory of Structures-I
236
12
2bM
bW
b
bM
bWpext ±=±= . . . (9.11)
We have 2
6bWe
bWpext ±= . . . (9.11a)
Masonry is assumed to be capable of taking compressive stresses but no tensile stresses. As tensile stresses will occur first at the point C (the point of smallest stress) if pc < 0, stress at C is tensile.
or 2
06 W . eW b b
± < for tensile stress.
or be61< . . . (9.12)
or writing it in the other way 6be > . . . . (9.12a)
In other words if the eccentricity is more than 6b to the left of O there will be
tensile stresses at O. (Similarly, if the eccentricity is more than 6b to the right of
O there will be tensile stresses at D.) Thus, we see that as long as the resultant lies within the middle third of the base there will not be any tensile stress in the section. This is the famous middle third rule. Example 9.4
A masonry retaining wall of trapezoidal section retains level earth 6 metres high. The retaining wall is 1 m wide at the top, determine the bottom width so that no tension is induced in the base. The unit weight of masonry is 23 kN/m3 and of soil 15 kN/m3. The angle of repose of the soil is 30o and the back face of the wall is vertical.
Figure 9.9
Find the value of the maximum pressure at the base. Solution
The horizontal pressure ph at any depth h is given by
6 m
A1 m
B
2 m
G
D
O′ D′
CeO
P
b/2b/2
W
Ex ph (max)
C′
225
Retaining Wallso
o1 sin 1 sin 30. 151 sin 1 sin 30
hp = w h h− φ −⋅ = ⋅
+ φ +
2115 5 (kN/m )3hp h h= × =
ph(max) = 5 × 6 = 30 kN/m2 (at h = 6 m) at base Considering 1 m width of the wall.
Total horizontal pressure 30 6 90 kN2
P ×= = acts at a height of 6 2 m
3=
from the bottom edge C. If the width of the base of the wall for no tension is ‘b’ then weight of 1 m
width of wall 123 1 62
b+⎛× × ×⎜⎝ ⎠
⎞⎟ kN = 69 (1 + b) kN.
It acts through x where x is the distance of CG of this trapezium ABCD from line BC.
D
A B
C
6 m CG
G2
1 m
x2
b
x 1 A2
A1
x
G1
1 1 2 2
1 2
A x A xx
A A+
=+
161 6 0.5 ( 1) 12 3
61 6 ( 1)2
bbx
b
−⎛ ⎞× × + − × × +⎜ ⎟⎝ ⎠=
× + − ×
2 13( 1)
b bb+ +
=+
The weight line GE meets the wall base CD; at E such that 2 21 2,3( 1) 3( 1)
b b b bEC DE b ECb b
2 1+ + += = − =
+ +−
Hence, eccentricity (e) of the weight, 2 22 2 13( 1) 2 6 ( 1)
b b b bbOE DE DOb b+ − + −
= − = − =+ +
2
The net bending moment at base, 3hM P W OE= ⋅ − ⋅
22290 2 69 (1 ) 180 11.5( 2)
6 (1 )b bb b
b+ −
= × − + = − + −+
b kNm
∴ Minimum bending stress at 3 2
621 1
12
bM MW WC b bb b
⋅= − = −
× ×
2
2 269 (1 ) 69 ( 2)1080b b
b b b
⎛ ⎞+ += − −⎜ ⎟⎜ ⎟
⎝ ⎠
b −
For no tension at base DC the above stress at C must be zero
∴ 2
2 269 (1 ) 69 ( 2)1080b b
b b b
+ += −
b −
giving, b (1 + b) = 15.652 – (b2 + b – 2) or, b2 + b – 8.826 = 0
226
Theory of Structures-I Giving Giving 1 6.025 2.513 m
2b − ±= = (taking the positive value)
Hence take the base width as 2.6 m. Maximum pressure at base at point D is
3
269 (1 2.513) 6 [180 (2.513 2.513 2) 11.5
2.513 2.513
+ × − + − ×= +
96.46 96.46 192.92= + = kN/m2
S AQ 1 AQ 1
(a) In Example 9.4, find the factor of safety against sliding and overturning if the coefficient of friction between wall base and soil is 0.5.
(b) A masonry retaining wall of trapezoidal section is 10 m high and has 1.5 m top width and 6.5 m bottom width. The earth face of the retaining wall has a better of 1 in 10, and the soil surface retained is level at the top of the wall. Find the maximum and minimum normal stresses at the base. Masonry weights 23 kN/m3 and soil 18 kN/m3. The angle of repose of the soil is 30o.
Figure 9.10
6.5 m
1.5 m A B E
C G F
1 m
10 m
D
4 m
6.5 m
227
Retaining Walls9.7 RETAINING WALLS WITH INCLINED EARTH
SLOPES
Sometimes the earth retained behind a retaining wall is not horizontal but sloping at an angle α with it (Figure 9.11). It has been proved that in such cases the pressure on the retaining wall at a depth x is given by
2 2
2 2
cos α cos α coscos α
cos α cos α cosxp = w . x
− −
+ −
φ
φ . . . (9.13)
where w = Density of the soil, and
φ = Angle of internal friction (or angle of repose) of soil.
Figure 9 .11
It acts at an angle α with the horizontal (i.e. parallel to the earth slope). The total pressure P acting on the surface BC of the wall will be
2 22
2 2
cos α cos α coscos α
2 cos α cos α cos
WhP⎡ ⎤− −⎢ ⎥=⎢ ⎥+ −⎣ ⎦
φ
φ . . . (9.14)
P acts at a height3h from the base DC of the retaining wall.
[Note : It can be seen that if we put α = 0 in Eqs. (9.13) and (9.14), it reduces to the formulae which are Eqs. (9.8) and (9.9) for the case of soil with horizontal surface.]
Example 9.5
A masonry retaining wall of trapezoidal section and earth side face vertical has a top width of 1.5 m and base width of 3.5 m. The wall is 6 m high and retains earth at a slope of 1 vertical to 2 horizontal. The weight of soil retained is 18 kN/m3 and the angle of repose is 30o. Find the maximum and minimum pressure intensities at the base of the wall. Weight of masonry is 23 kN/m3.
A B
1 2 α
1.5 m
6 m
CD
Pv P
Ph
3.5 m
W
α
x
h
A B
CD
PvPPh
α
α
228
Theory of Structures-I
(a) Retaining Wall
108.
291
(b) Pressure Diagram Figure 9.12
Solution
Considering 1 m width of wall (Figure 9.12).
1tan α2
=
∴ α = 26.57o and cos α = 0.8944
φ = 30o ∴ cos φ = 0.866
∴ Coefficient of active earth pressure
2 2
2 2
cos cos coscos
cos cos cosaK
α − α −= α
φ
α + α − φ
= 0.5366
∴ 2 260.5366 18 173.87
2 2aWhP = K = × × = kN
which is inclined at an angle α = 26.57o with horizontal.
Horizontal component of P, Ph = P cos α = 173.87 cos 26.57o = 155.5 kN
Vertical component of P; Pv = P sin α = 173.87 sin 26.57o =77.7 kN
and P acts at a distance 6 2 m3 3h= = above the base.
Weight of masonry wall (per m width),
1.5 3.5ρ 6 23 345 kN2 2
a + bW h += ⋅ ⋅ = × × =
This acts at a distance of x from C,
where 2 2
1.317 m3( )
a ab + bxa + b
+= =
∴ Total pressure acting at point C of the base will be
133.
33
D′ C′
229
Retaining Walls
2 2
6 622 2v
v hc
b bW x PP PWp = +
b b b b
⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ + −
2× h
b
2 26 345(1.75 1.317)345 77.7 6 77.7 1.75 2 155.5 6
3.5 3.5 (3.5) (3.5) (3.5)2× − × × ×
= + + + −×
= 108.21 kN/m2
And total pressure acting at point D of the base will be
2 2
6 6( )2 .2 2
D
vv h
b bW x PP PWp = +
b b b b
⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠− − +
2h
b
= 133.33 kN/m2
9.8 MINIMUM DEPTH OF FOUNDATION
Consider a column carrying a load P to be transmitted to the soil through a footing of area (A) as in Figure 9.13.
p1
p2M
P3
N
P
h
Figure 9.13
Hence, the vertical pressure on soil just below the footing is
APp =1 . . . (9.15)
So on a soil element ‘M’ just below the footing the horizontal pressure p2 caused by the footing is
2 11 sin1 sin
p p − φ=
+ φ . . . (9.16)
On a wedge ‘N’ just outside the footing this horizontal pressure p2 will cause a vertical pressure p3 such that
3 21 sin1 sin
p p − φ=
+ φ . . . (9.17)
Substituting Eq. (9.16) in Eq. (9.17), we get
3 1
2 21 sin 1 sin1 sin 1 sin
Pp pA
⎛ ⎞ ⎛ ⎞− φ − φ= =⎜ ⎟ ⎜ ⎟+ φ + φ⎝ ⎠ ⎝ ⎠
We know that the vertical pressure p3 = wh, where w is the unit weight of soil and h is the depth of the footing.
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Theory of Structures-I
∴ 21 sin
1 sinPwh =A⎛ ⎞− φ⎜ ⎟+ φ⎝ ⎠
or the minimum depth of footing for equilibrium will be 21 sin
1 sinPh =
wA⎛ ⎞− φ⎜ ⎟+ φ⎝ ⎠
SAQ 2
A column carries a load of 1200 kN on a masonry footing which is 2 × 2 m in area. Find the minimum foundation depth necessary if the unit weight of soil is 18 kN/m3 and has an angle of repose of 35o (assume self weight of footing as 10% of the column load).
9.9 SUMMARY
A wall used to retain some material on one or both sides of it can be termed as retaining wall. Generally, it is used to retain soil at two different levels on either side of the wall. Moreover, the materials to be retained on either side may be different, e.g. the wall of a swimming pool retains soil on one side and water on the other side. When the material retained by it is a fluid say water, it is called a dam.
Gravity wall, cantilever wall, and buttress wall are some of the common types of retaining walls. In case of Gravity Wall, the self-weight of the structure provides stability against the pressure of retained earth. Cantilever wall consists of the vertical arm, which retains the earth and is held in position by the base slab. In this case, the weight of fill on top of the heel, in addition to the weight of the wall, contributes to the stability of the structure.
The process of design of a retaining wall commences with preliminary proportioning of the wall, and then the design is checked against the stability requirements of wall and is revised if required.
9.10 ANSWERS TO SAQs
SAQ 1
(a) (i) Maximum vertical reaction = 242.4 kN
∴ Frictional force preventing sliding = 121.2 kN
∴ Factor of safety against sliding 242.4 1.35121.2
= .
(ii) Maximum overturning moment about toe = 180 kN.
Maximum stabilizing moment about the same point = 191.55 kNm.
∴ Factor of safety against overturning 191.55 1.06180
= .
(b) From Figure 9.14 we have (considering 1 m length of wall)
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Retaining Walls
11 4 10 23 460 kN2
W = × × × =
2 1.5 10 23 345 kNW = × × =
31 1 10 23 115 kN2
W = × × × =
41 10 1 18 90 kN2
W = × × × =
14 232.5 m3 6
x = + =
21.5 71.0 m2 4
x = + =
32 m3
x =
41 m3
x =
1010 kNW∑ =
6.5 m
10 m
10/3 m
1.5 m A B
C D
O F
4 m
W1
W2
W3
W4x4
X2
X1
X3
6.5 m
1 m
Earth Pressure
x3 = 2.45 m e = 0.8 3.25 m
2 1 sin2 1 sinH
wHP⎛ ⎞− φ
= ⎜ ⎟+ φ⎝ ⎠
(a) Location of Retaining Wall
182 kN/m2
128 kN/m2
(b) Foundation Pressure Diagram
Figure 9.14
∴
2 7 2 1460 345 115 903 4 3 3 2.45 m
460 345 115 90i i
i
W xx
W
× + × + × + ×∑= = =
∑ + + +
6.5 2.45 3.25 2.45 0.8 m2
e = − = − =
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Theory of Structures-I
2 2 o
o1 sin 18 10 1 sin 30 = 300 kN
2 1 sin 2 1 sin 30H
W HP⎛ ⎞− φ × −
= = ×⎜ ⎟+ φ +⎝ ⎠
10 m3
h =
∴ 10300 1010 0.8 192 kN/m3
M P h W e= − = × − × =
∴ Pressure below base are
At toe, 22 2
6 6 1921010 182 kN/m6.5 (6.5)
DMWp
b b
×= + = + =
At heel, 22 2
6 6 1921010 128 kN/m6.5 (6.5)
CMWp
b b
×= − = − =
SAQ 2
Allowing 10% for self weight of footing.
Minimum depth of foundation, 21 sin
1 sinPh =
wA⎛ ⎞− φ⎜ ⎟+ φ⎝ ⎠
2o
o1200 1.1 1 sin 35 1.35 m
18 (2 2) 1 sin 35h =
⎛ ⎞× −=⎜ ⎟⎜ ⎟× × +⎝ ⎠
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Retaining Walls
FURTHER READING Gupta S. P., Pandit, G. S., Gupta, R. (1999), Theory of Structures, Vol. I and II, Tata McGraw-Hill Publishing Company Limited, New Delhi.
Negi, L. S. and Jangid, R. S. (1997), Structural Analysis, Tata McGraw-Hill Publishing Company Limited, New Delhi.
Punmia, B. C., Jain, Ashok and Jain, Arun (1998), Design of Steel Structures, Laxmi Publications, New Delhi.
Punmia, B. C., Jain, Ashok and Jain, Arun (2004), Theory of Structures, Laxmi Publications, New Delhi.
Ramchandra, Dr., Ratwani, M. M. (1991), Design of Steel Structures, Standard Book House, New Delhi.
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THEORY OF STRUCTURES-I In any construction project, all components are designed properly before the execution of the project. For designing the structural components, they are being analysed first to calculate the bending moment, shear force and their variations on the basis of given load pattern. This forms the subject matter of this course. In this course, you are introduced to the concepts of analysis and design of structures.
This course consists of nine units.
The first three units discuss some of the standard methods of analysis of structural components. The next five units explain the method of structural design as applied to steel structures. The last unit describes the design of retaining structures, which retain the load of earthfill or water (in a reservoir).
Unit 1 deals with the analysis of moving load systems, and the use of influence line diagrams. The concept of influence lines is very useful in design of all such structures which carry moving loads like bridges, flyovers, crane girder, etc.
Unit 2 explains the methods of analysis of statically indeterminate structures, i.e. structures which cannot be analysed by the principles of statics only which you have learnt in earlier courses, namely Applied Mechanics and Strength of Materials.
Unit 3 introduces you to the powerful tool of Moment Distribution Method, which is quite convenient for the analysis of some statically indeterminate structures. Towards the end, the unit also explains portal frames.
Unit 4 deals with the design of joints, which are mainly comprise riveted or welded joints.
Unit 5 deals with the design of steel members subjected to tensile load only. It explains the loads and stresses considered for the design of tension members.
Unit 6 explains the design of steel members subjected to compressive forces, e.g. columns and struts. While explaining permissible stresses in compression members, the unit presents the design procedure for different types of compression member including compound compression members.
Unit 7 presents the methods for design of steel members subjected to bending (flexural) forces, e.g. beams, girders, purlins, etc. The unit explains the classical theory of bending.
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In Unit 8, you will find the design of steel roof trusses, which is one of the most common forms of structure, normally employed to cover large spaces.
Theory of Structures-I
Finally, Unit 9 discusses the designs of retaining walls and dams, which are mainly massive structures used to sustain large horizontal forces due to the retained materials (earth or water).
The Self-Assessment Questions (SAQs), given in each unit, are intended to help you in checking your own progress. You should study the text carefully, and then try to solve the SAQs and verify your answers with those given at the end of each unit. This will definitely develop your confidence.
At the end, we wish you all the best for your all future educational endeavours.